Rooted complete minors in line graphs with a Kempe coloring

TL;DR

This paper proves a conjecture related to Kempe colorings in line graphs, showing that certain vertex colorings guarantee the existence of a complete minor with specific properties.

Contribution

It establishes the conjecture for line graphs, demonstrating a new connection between Kempe colorings and complete minors in this class of graphs.

Findings

Proved the conjecture for line graphs.

Connected vertex colorings imply existence of specific complete minors.

Advances understanding of graph minors in relation to Kempe colorings.

Abstract

It has been conjectured that if a finite graph has a vertex coloring such that the union of any two color classes induces a connected graph, then for every set $T$ of vertices containing exactly one member from each color class there exists a complete minor such that $T$ contains exactly one member from each branching set. Here we prove the statement for line graphs.